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Theorem ordtypelem10 7737
Description: Lemma for ordtype 7742. Using ax-rep 4400, exclude the possibility that  O is a proper class and does not enumerate all of 
A. (Contributed by Mario Carneiro, 25-Jun-2015.)
Hypotheses
Ref Expression
ordtypelem.1  |-  F  = recs ( G )
ordtypelem.2  |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }
ordtypelem.3  |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C  A. u  e.  C  -.  u R v ) )
ordtypelem.5  |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }
ordtypelem.6  |-  O  = OrdIso
( R ,  A
)
ordtypelem.7  |-  ( ph  ->  R  We  A )
ordtypelem.8  |-  ( ph  ->  R Se  A )
Assertion
Ref Expression
ordtypelem10  |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  A
) )
Distinct variable groups:    v, u, C    h, j, t, u, v, w, x, z, R    A, h, j, t, u, v, w, x, z    t, O, u, v, x    ph, t, x    h, F, j, t, u, v, w, x, z
Allowed substitution hints:    ph( z, w, v, u, h, j)    C( x, z, w, t, h, j)    T( x, z, w, v, u, t, h, j)    G( x, z, w, v, u, t, h, j)    O( z, w, h, j)

Proof of Theorem ordtypelem10
Dummy variables  b 
c  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordtypelem.1 . . 3  |-  F  = recs ( G )
2 ordtypelem.2 . . 3  |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }
3 ordtypelem.3 . . 3  |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C  A. u  e.  C  -.  u R v ) )
4 ordtypelem.5 . . 3  |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }
5 ordtypelem.6 . . 3  |-  O  = OrdIso
( R ,  A
)
6 ordtypelem.7 . . 3  |-  ( ph  ->  R  We  A )
7 ordtypelem.8 . . 3  |-  ( ph  ->  R Se  A )
81, 2, 3, 4, 5, 6, 7ordtypelem8 7735 . 2  |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  ran  O ) )
91, 2, 3, 4, 5, 6, 7ordtypelem4 7731 . . . . 5  |-  ( ph  ->  O : ( T  i^i  dom  F ) --> A )
10 frn 5562 . . . . 5  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  ran  O  C_  A )
119, 10syl 16 . . . 4  |-  ( ph  ->  ran  O  C_  A
)
12 simprl 750 . . . . . . . . 9  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  b  e.  A
)
136adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  R  We  A
)
147adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  R Se  A )
151, 2, 3, 4, 5, 13, 14ordtypelem8 7735 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O  Isom  _E  ,  R  ( dom  O ,  ran  O ) )
16 isof1o 6013 . . . . . . . . . . . . 13  |-  ( O 
Isom  _E  ,  R  ( dom  O ,  ran  O )  ->  O : dom  O -1-1-onto-> ran  O )
17 f1of 5638 . . . . . . . . . . . . 13  |-  ( O : dom  O -1-1-onto-> ran  O  ->  O : dom  O --> ran  O )
1815, 16, 173syl 20 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O : dom  O --> ran  O )
19 f1of1 5637 . . . . . . . . . . . . . 14  |-  ( O : dom  O -1-1-onto-> ran  O  ->  O : dom  O -1-1-> ran 
O )
2015, 16, 193syl 20 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O : dom  O
-1-1-> ran  O )
21 simpl 454 . . . . . . . . . . . . . . 15  |-  ( ( b  e.  A  /\  -.  b  e.  ran  O )  ->  b  e.  A )
22 seex 4679 . . . . . . . . . . . . . . 15  |-  ( ( R Se  A  /\  b  e.  A )  ->  { c  e.  A  |  c R b }  e.  _V )
237, 21, 22syl2an 474 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  { c  e.  A  |  c R b }  e.  _V )
2411adantr 462 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ran  O  C_  A
)
25 rexnal 2724 . . . . . . . . . . . . . . . . . . 19  |-  ( E. m  e.  dom  O  -.  ( O `  m
) R b  <->  -.  A. m  e.  dom  O ( O `
 m ) R b )
261, 2, 3, 4, 5, 6, 7ordtypelem7 7734 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  b  e.  A )  /\  m  e.  dom  O )  -> 
( ( O `  m ) R b  \/  b  e.  ran  O ) )
2726ord 377 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  b  e.  A )  /\  m  e.  dom  O )  -> 
( -.  ( O `
 m ) R b  ->  b  e.  ran  O ) )
2827rexlimdva 2839 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  b  e.  A )  ->  ( E. m  e.  dom  O  -.  ( O `  m ) R b  ->  b  e.  ran  O ) )
2925, 28syl5bir 218 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  b  e.  A )  ->  ( -.  A. m  e.  dom  O ( O `  m
) R b  -> 
b  e.  ran  O
) )
3029con1d 124 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  b  e.  A )  ->  ( -.  b  e.  ran  O  ->  A. m  e.  dom  O ( O `  m
) R b ) )
3130impr 616 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  A. m  e.  dom  O ( O `  m
) R b )
32 ffun 5558 . . . . . . . . . . . . . . . . . . . 20  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  Fun  O )
339, 32syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  Fun  O )
34 funfn 5444 . . . . . . . . . . . . . . . . . . 19  |-  ( Fun 
O  <->  O  Fn  dom  O )
3533, 34sylib 196 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  O  Fn  dom  O
)
3635adantr 462 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O  Fn  dom  O )
37 breq1 4292 . . . . . . . . . . . . . . . . . 18  |-  ( c  =  ( O `  m )  ->  (
c R b  <->  ( O `  m ) R b ) )
3837ralrn 5843 . . . . . . . . . . . . . . . . 17  |-  ( O  Fn  dom  O  -> 
( A. c  e. 
ran  O  c R
b  <->  A. m  e.  dom  O ( O `  m
) R b ) )
3936, 38syl 16 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ( A. c  e.  ran  O  c R b  <->  A. m  e.  dom  O ( O `  m
) R b ) )
4031, 39mpbird 232 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  A. c  e.  ran  O  c R b )
41 ssrab 3427 . . . . . . . . . . . . . . 15  |-  ( ran 
O  C_  { c  e.  A  |  c R b }  <->  ( ran  O 
C_  A  /\  A. c  e.  ran  O  c R b ) )
4224, 40, 41sylanbrc 659 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ran  O  C_  { c  e.  A  |  c R b } )
4323, 42ssexd 4436 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ran  O  e.  _V )
44 f1dmex 6546 . . . . . . . . . . . . 13  |-  ( ( O : dom  O -1-1-> ran 
O  /\  ran  O  e. 
_V )  ->  dom  O  e.  _V )
4520, 43, 44syl2anc 656 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  dom  O  e.  _V )
46 fex 5947 . . . . . . . . . . . 12  |-  ( ( O : dom  O --> ran  O  /\  dom  O  e.  _V )  ->  O  e.  _V )
4718, 45, 46syl2anc 656 . . . . . . . . . . 11  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O  e.  _V )
481, 2, 3, 4, 5, 13, 14, 47ordtypelem9 7736 . . . . . . . . . 10  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O  Isom  _E  ,  R  ( dom  O ,  A ) )
49 isof1o 6013 . . . . . . . . . 10  |-  ( O 
Isom  _E  ,  R  ( dom  O ,  A
)  ->  O : dom  O -1-1-onto-> A )
50 f1ofo 5645 . . . . . . . . . 10  |-  ( O : dom  O -1-1-onto-> A  ->  O : dom  O -onto-> A
)
51 forn 5620 . . . . . . . . . 10  |-  ( O : dom  O -onto-> A  ->  ran  O  =  A )
5248, 49, 50, 514syl 21 . . . . . . . . 9  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ran  O  =  A )
5312, 52eleqtrrd 2518 . . . . . . . 8  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  b  e.  ran  O )
5453expr 612 . . . . . . 7  |-  ( (
ph  /\  b  e.  A )  ->  ( -.  b  e.  ran  O  ->  b  e.  ran  O ) )
5554pm2.18d 111 . . . . . 6  |-  ( (
ph  /\  b  e.  A )  ->  b  e.  ran  O )
5655ex 434 . . . . 5  |-  ( ph  ->  ( b  e.  A  ->  b  e.  ran  O
) )
5756ssrdv 3359 . . . 4  |-  ( ph  ->  A  C_  ran  O )
5811, 57eqssd 3370 . . 3  |-  ( ph  ->  ran  O  =  A )
59 isoeq5 6011 . . 3  |-  ( ran 
O  =  A  -> 
( O  Isom  _E  ,  R  ( dom  O ,  ran  O )  <->  O  Isom  _E  ,  R  ( dom 
O ,  A ) ) )
6058, 59syl 16 . 2  |-  ( ph  ->  ( O  Isom  _E  ,  R  ( dom  O ,  ran  O )  <->  O  Isom  _E  ,  R  ( dom 
O ,  A ) ) )
618, 60mpbid 210 1  |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  A
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   E.wrex 2714   {crab 2717   _Vcvv 2970    i^i cin 3324    C_ wss 3325   class class class wbr 4289    e. cmpt 4347    _E cep 4626   Se wse 4673    We wwe 4674   Oncon0 4715   dom cdm 4836   ran crn 4837   "cima 4839   Fun wfun 5409    Fn wfn 5410   -->wf 5411   -1-1->wf1 5412   -onto->wfo 5413   -1-1-onto->wf1o 5414   ` cfv 5415    Isom wiso 5416   iota_crio 6048  recscrecs 6827  OrdIsocoi 7719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-recs 6828  df-oi 7720
This theorem is referenced by:  ordtype  7742
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